Q93SE A rock specimen from a particula... [FREE SOLUTION]

91影视

Probability And Statistics For Engineering And Sciences

Jay L. Devore

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 714 pages

ISBN: 9781305251809

Chapter 5: Q93SE (page 245)

A rock specimen from a particular area is randomly selected and weighed two different times. Let ${\rm{W}}$denote the actual weight and ${{\rm{X}}_{\rm{1}}}$and ${{\rm{X}}_{\rm{2}}}$the two measured weights. Then ${{\rm{X}}_{\rm{1}}}{\rm{ = W + }}{{\rm{E}}_{\rm{1}}}$and${{\rm{X}}_{\rm{2}}}{\rm{ = W + }}{{\rm{E}}_{\rm{2}}}$, where ${{\rm{E}}_{\rm{1}}}$and ${{\rm{E}}_{\rm{2}}}$are the two measurement errors. Suppose that the ${{\rm{E}}_{\rm{i}}}$ 's are independent of one another and of ${\rm{W}}$and that${\rm{V}}\left( {{{\rm{E}}_{\rm{1}}}} \right){\rm{ = V}}\left( {{{\rm{E}}_{\rm{2}}}} \right){\rm{ = \sigma }}_{{{\rm{E}}^{\rm{2}}}}^{\rm{2}}$.
a. Express${\rm{\rho }}$, the correlation coefficient between the two measured weights ${{\rm{X}}_{\rm{1}}}$and${{\rm{X}}_{\rm{2}}}$, in terms of${\rm{\sigma }}_{\rm{X}}^{\rm{2}}$, the variance of actual weight, and${\rm{\sigma }}_{\rm{X}}^{\rm{2}}$, the variance of measured weight.
b. Compute ${\rm{\rho }}$when ${{\rm{\sigma }}_{\rm{W}}}{\rm{ = 1\;kg}}$ and ${{\rm{\sigma }}_{\rm{E}}}{\rm{ = }}{\rm{.01\;kg}}{\rm{.}}$

Short Answer

Expert verified

a) The correlation coefficient ${{\rm{\rho }}_{{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}}}{\rm{ = }}\frac{{{\rm{\sigma }}_{\rm{W}}^{\rm{2}}}}{{{\rm{\sigma }}_{\rm{W}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{E}}^{\rm{2}}}}$

b) The correlation coefficient ${{\rm{\rho }}_{{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}}}{\rm{ = 0}}{\rm{.9999}}$

Step by step solution

Definition

The standard deviation is a measurement of a collection of values' variance or dispersion. A low standard deviation implies that the values are close to the set's mean, whereas a high standard deviation suggests that the values are dispersed over a larger range.

Calculating the correlation coefficient

(a):

correlation coefficient

of ${\rm{X}}$ and ${\rm{Y}}$ is

${{\rm{\rho }}_{{\rm{X,Y}}}}{\rm{ = }}\frac{{{\rm{Cov(X,Y)}}}}{{{{\rm{\sigma }}_{\rm{X}}}{\rm{ \times }}{{\rm{\sigma }}_{\rm{Y}}}}}$

The variances of random variables ${{\rm{X}}_{\rm{1}}}$ and ${{\rm{X}}_{\rm{2}}}$are

$\begin{aligned}{\rm{V}}\left( {{{\rm{X}}_{\rm{1}}}} \right)&= V\left( {{\rm{W + }}{{\rm{E}}_{\rm{1}}}} \right)\\ &= V(W) + V \left( {{{\rm{E}}_{\rm{1}}}} \right)\\&= \sigma _{\rm{W}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{E}}^{\rm{2}}{\rm{V}}\left( {{{\rm{X}}_{\rm{2}}}} \right)\\&= V \left( {{\rm{W + }}{{\rm{E}}_{\rm{2}}}} \right)\\&= V(W) + V\left( {{{\rm{E}}_{\rm{1}}}} \right)\\&= \sigma _{\rm{W}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{E}}^{\rm{2}}\end{aligned}$

and the covariance is

$\begin{aligned}{\rm{Cov}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right)& = Cov\left( {{\rm{W + }}{{\rm{E}}_{\rm{1}}}{\rm{,W + }}{{\rm{E}}_{\rm{2}}}} \right)\\&= Cov(W,W) + Cov\left( {{\rm{W,}}{{\rm{E}}_{\rm{2}}}} \right){\rm{ + Cov}}\left( {{{\rm{E}}_{\rm{1}}}{\rm{,W}}} \right){\rm{ + Cov}}\left( {{{\rm{E}}_{\rm{1}}}{\rm{,}}{{\rm{E}}_{\rm{2}}}} \right)\\&= Cov(W,W) + 0 + 0 + 0\\&= \sigma _{\rm{W}}^{\rm{2}}{\rm{n}}\end{aligned}$

Therefore, the correlation coefficient

$\begin{array}{c}{{\rm{\rho }}_{{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}}}{\rm{ = }}\frac{{{\rm{Cov}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right)}}{{{{\rm{\sigma }}_{{{\rm{X}}_{\rm{1}}}}}{\rm{ \times }}{{\rm{\sigma }}_{{{\rm{X}}_{\rm{2}}}}}}}\\{\rm{ = }}\frac{{{\rm{\sigma }}_{\rm{W}}^{\rm{2}}}}{{\sqrt {{\rm{\sigma }}_{\rm{W}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{E}}^{\rm{2}}} \sqrt {{\rm{\sigma }}_{\rm{W}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{E}}^{\rm{2}}} }}\\{\rm{ = }}\frac{{{\rm{\sigma }}_{\rm{W}}^{\rm{2}}}}{{{\rm{\sigma }}_{\rm{W}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{E}}^{\rm{2}}}}.\end{array}$

Calculating the correlation coefficient

For${{\rm{\sigma }}_{\rm{W}}}{\rm{ = 1 and }}{{\rm{\sigma }}_{\rm{E}}}{\rm{ = 0}}{\rm{.01}}$the following holds

$\begin{array}{c}{{\rm{\rho }}_{{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}}}{\rm{ = }}\frac{{{\rm{\sigma }}_{\rm{W}}^{\rm{2}}}}{{{\rm{\sigma }}_{\rm{W}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{E}}^{\rm{2}}}}\\{\rm{ = }}\frac{{{{\rm{1}}^{\rm{2}}}}}{{{{\rm{1}}^{\rm{2}}}{\rm{ + 0}}{\rm{.0}}{{\rm{1}}^{\rm{2}}}}}\\{\rm{ = 0}}{\rm{.9999}}{\rm{.}}\end{array}$